Pólya Web: Coalescing Random Walks on ℕ²

• Pólya Web is a system of coalescing random walks on ℕ² where each walk follows state-dependent probabilities based on its current coordinates.
• It establishes key connections between local bias, negative association via the BKR inequality, and the strong law showing cluster count scaling as approximately √(πn).
• The model’s edge-scaling limit transitions to the Yule Web, where discrete Pólya walks converge to continuum processes with Gamma-distributed martingale limits.

The Pólya Web is a system of coalescing random walks founded on the classical Pólya urn process, situated on the integer lattice $\mathbb{N}^2$. It serves as an analogue to the up-right oriented web of coalescing random walks introduced by Tóth and Werner (1998), but replaces simple symmetric random walks with Pólya walks as the fundamental constituents. The Pólya Web reveals deep connections between the geometry of random walks with state-dependent bias, negative dependence structures, strong laws for clustering phenomena, determinantal formulas for joint limit distributions, and continuum scaling limits leading to the Yule Web of coalescing Poissonian processes (Urbán, 17 Jan 2026).

1. Construction of the Pólya Web

A single Pólya walk is defined via an inhomogeneous Markov process on %%%%1%%%%. Given initial state $(X_0, Y_0) = (x_0, y_0)$, the transitions are

$$P\bigl((X_{n+1}, Y_{n+1}) = (x+1, y)\mid (X_n, Y_n) = (x, y)\bigr) = \frac{x}{x+y},$$

$$P\bigl((X_{n+1}, Y_{n+1}) = (x, y+1)\mid (X_n, Y_n) = (x, y)\bigr) = \frac{y}{x+y},$$

where the walk increment is either in the $x$-direction or $y$-direction with probability proportional to the current coordinate values.

A lattice-arrow representation is available: assign to each site $\lambda = (k, \ell)$ an independent arrow $\omega_\lambda \in \{(1,0), (0,1)\}$, with

$$P(\omega_\lambda = (1,0)) = \frac{k}{k+\ell}, \qquad P(\omega_\lambda = (0,1)) = \frac{\ell}{k+\ell}.$$

Given $\lambda \in \mathbb{N}^2$, the walk $S_\lambda(n)$ born at time $n = k + \ell$ starts at $\lambda$ and follows the arrows: $$S_\lambda(n+1) = S_\lambda(n) + \omega_{S_\lambda(n)}$$.

In the Pólya Web, a Pólya walk is initiated at every lattice site. Trajectories that intersect at any site coalesce and continue together thereafter, forming an up-right web of coalescing Pólya walks.

2. Negative Association and the van den Berg–Kesten–Reimer Inequality

Consider a finite region $\Lambda_N = \{\lambda: \|\lambda\| \leq N\}$ and product space $\Omega_N = \{(1,0), (0,1)\}^{\Lambda_N}$ under the natural product measure. For two neighboring start-points $\lambda_{k-1} = (k-1, n-k+1)$ and $\lambda_k = (k, n-k)$ at level $n$, define the indicator

$$I_k(N) = \mathbf{1}\{\text{walks from }\lambda_{k-1}, \lambda_k \text{ do not coalesce by time }N\}.$$

As $N \to \infty$, $I_k(N) \uparrow I_k = \mathbf{1}\{$never coalesce$\}$.

The geometry of the dual web ensures that “no-meet” events for disjoint sets $A, B$ of indices are certified on disjoint subsets of arrow variables, enabling the use of the van den Berg–Kesten–Reimer (BKR) inequality: $$\mathbb{P}_N(E \,\square\, F) \leq \mathbb{P}_N(E)\mathbb{P}_N(F),$$ for any two events $E, F \subset \Omega_N$.

In the current context, for disjoint sets $A, B$,

$$\mathbb{P}_N(\{I_j(N)=1: j\in A\} \cap \{I_j(N)=1: j\in B\}) \leq \mathbb{P}_N(\{I_j(N)=1: j\in A\}) \mathbb{P}_N(\{I_j(N)=1: j\in B\}).$$

Letting $N \to \infty$ yields negative association for the family $\{I_1, \dots, I_n\}$: for any two increasing functions $f, g$ of disjoint subsets,

$$E[f(I_j: j\in A)\, g(I_j: j\in B)] \le E[f(I_j: j\in A)]\, E[g(I_j: j\in B)].$$

This structure is pivotal in establishing rigorous large deviations, variance bounds, and limit laws.

3. Strong Law for the Number of Clusters

For walks started at level $n$, coalescence partitions the $n+1$ start-points $\lambda_k = (k, n-k)$ into clusters, with $C_n$ denoting the total number. One has

$C_n = 1 + \sum_{k=1}^n I_k,$

where $I_k$ indicates that $\lambda_{k-1}$ and $\lambda_k$ never meet.

Using the joint law of limiting Beta variables (see Section 4), it is shown that

$$E[C_n] \sim \sqrt{\pi n}, \qquad \operatorname{Var}(C_n) = O(\sqrt{n}).$$

Large deviations for negatively associated $0$-$1$ variables (by Dubhashi–Ranjan 1998) imply the exponentially small probability of significant deviation from the mean.

Consequently, by the strong law of large numbers,

$$\frac{C_n}{E[C_n]} \to 1 \quad \text{a.s. as } n \to \infty,$$

and thus almost surely $C_n \sim \sqrt{\pi n}$.

4. Joint Limiting Law for Normalized Coordinates

For a Pólya walk started at $\lambda = (a, b)$, the classical martingale gives

$$Z_\lambda(n) = \frac{X_\lambda(n)}{X_\lambda(n) + Y_\lambda(n)} = \frac{X_\lambda(n)}{n} \to Z_\lambda \sim \operatorname{Beta}(a, b) \qquad \text{a.s.}$$

For points related by $\lambda_1 \succeq \lambda_2$ (i.e., $x_1 \le x_2, y_1 \ge y_2$): $$Z_{\lambda_1}(n) \le Z_{\lambda_2}(n) \implies Z_{\lambda_1} \le Z_{\lambda_2} \quad \text{a.s.}$$

The joint density of ordered limits $$\{Z_{\lambda_1} = Z_{r_1} = x_1 < \dots < Z_{\lambda_n} = Z_{r_n} = x_n\}$$ admits a determinantal formula. Let $f_u(x)$ and $F_u(x)$ be the marginal Beta density and CDF for parameter $u$: $$\frac{d}{dx_1 \cdots dx_n}\, P[Z_{\lambda_i} = Z_{r_i} = x_i,\, i = 1\dots n] = \det \begin{pmatrix} 1 - F_{\lambda_1}(x_1) & f_{\lambda_1}(x_1) & \cdots & 1 - F_{\lambda_1}(x_n) & f_{\lambda_1}(x_n) \ -F_{r_1}(x_1) & f_{r_1}(x_1) & \cdots & 1 - F_{r_1}(x_n) & f_{r_1}(x_n) \ \vdots & & \ddots & & \vdots \ -F_{r_n}(x_1) & f_{r_n}(x_1) & \cdots & -F_{r_n}(x_n) & f_{r_n}(x_n) \end{pmatrix}$$ The derivation applies the Karlin–McGregor determinant for noncrossing walks to the unnormalized coordinate process $X_\lambda(N)$, divides by $N^n$, and lets $N \to \infty$, incorporating coalescence via inclusion-exclusion on neighbors.

For two neighbors $\lambda_{k-1},\lambda_k$ at level $n$,

$$P(Z_{\lambda_{k-1}} < Z_{\lambda_k}) = P(\text{never meet}) = \frac{\binom{n-1}{k-1}^2}{\binom{2n-2}{2k-2}}.$$

This exact formula recovers $E[C_n] \sim \sqrt{\pi n}$.

5. Edge-Scaling Limit and the Yule Web

At fixed level $k \in \mathbb{N}$ and interval $[s,t] \subset (0, \infty)$, consider

$$J_n(k; s,t) = \#\{ \lambda = (j,k): j \in [ns, nt],\, \omega_\lambda = (0,1) \}$$

the number of up-steps from $[ns, nt]\times\{k\}$. $J_n$ is a sum of independent Bernoulli random variables with success probability $k/(j+k)$. The sum converges in distribution to a Poisson: $$J_n(k; s, t) \xrightarrow{d} \operatorname{Pois}\left(\int_s^t \frac{k}{r}\, dr\right).$$

The logarithmic change of variables $u = \ln r$ renders the rate $k/r$ constant over the continuum, suggesting a scaling window to real-valued levels. In this scaling limit, replace each discrete level $\{(j, k)\}$ by $\{(u, k): u \in \mathbb{R}\}$ and place a Poisson process of rate $k$ along $u$. The Yule walk $V_{k,s}(t)$ at $(k,s)$ proceeds to level $k+1$ at the next Poisson point to the right; coalescence is inherited from the discrete Pólya Web.

In the Yule Web, the analogue of the Beta limit is given by the martingale

$U_{k,s}(t) = e^{-(t-s)} V_{k,s}(t),$

with

$$U_{k,s}(t) \xrightarrow{a.s.} U_k, \qquad U_k \sim \Gamma(k,1).$$

The Yule Web maintains negative association for “no-meet” indicators as in the discrete case. The number of clusters among $n$ edge-starts in a window of width $t-s$ satisfies

$$C_n^{\mathrm{edge}} \sim \frac{t-s}{\sqrt{\pi}} \sqrt{n}, \quad \text{a.s.}$$

Joint distributions of the limiting Gamma variables are again given by a $2n \times 2n$ determinant, analogous to the Beta-case, but constructed from the Gamma densities $g_k(x) = x^{k-1}e^{-x}/\Gamma(k)$ and cumulative functions $G_k(x)$.

6. Summary of Key Properties and Connections

1. The Pólya walk is a directed, inhomogeneous random walk on $\mathbb{N}^2$, with local bias determined by site coordinates as $x/(x+y)$.
2. The coalescing system forms an up-right web where “no-meet” indicators for disjoint pairs of neighbors are negatively associated, owing to geometric separation in the dual web structure and the BKR–Reimer inequality.
3. The total number of clusters for walks born at a fixed level $n$ satisfies a strong law: $C_n/E[C_n]\to 1$ almost surely, and $E[C_n]\sim \sqrt{\pi n}$.
4. The joint law of normalized limiting coordinates $Z_\lambda$ for finite systems is given by a determinantal formula using the Karlin–McGregor method and inclusion-exclusion to account for coalescence.
5. Scaling to the edge by considering Poisson process approximations leads to the Yule Web. In this regime, normalized coordinate martingales converge almost surely to independent Gamma$(k,1)$ distributions, and joint laws are furnished by parallel determinantal constructions.

Further details, derivations, and all critical formulas are provided systematically in (Urbán, 17 Jan 2026).